I get 45p. Keep in mind that nanny needs a car maintained to a suitable standard, you don't want them driving your children around in a rust bucket. They need suitable insurance. They will be driving around during their own non-mileage paid time with equipment for work, such as car seats, possibly other things as well, so are carrying extra weight they would not need to carry if they didn't transport your children.
Whilst fuel may be 12p per mile, it's the cost of maintenance, insurance, and buying the car itself which are the big costs I feel. If you look at it that way, does 45p cover it?
See how it goes. All trips must be logged for mileage claim, so you can see over time what trips are made and if necessary put restrictions on where nanny goes.
On your budget, what figure have you estimated for annual mileage? Depending on your location it could well be £1000+
Mileage up to 10,000 miles per year is non taxable, as long as it is paid at 45p or less.
If you decide to pay less, make that clear now and be prepared for the nanny not to accept the job.